Finite rank deep kernel learning for robust time series forecasting and regression

ABSTRACT

Certain aspects of the present disclosure provide techniques for performing finite rank deep kernel learning. In one example, a method for performing finite rank deep kernel learning includes receiving a training dataset; forming a set of embeddings by subjecting the training data set to a deep neural network; forming, from the set of embeddings, a plurality of dot kernels; combining the plurality of dot kernels to form a composite kernel for a Gaussian process; receiving live data from an application; and predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel.

CROSS-REFERENCE TO RELATED APPLICATIONS

This Applications claims the benefit of U.S. Provisional Application No.62/755,229, filed on Nov. 2, 2018, the contents of which areincorporated herein in their entirety.

INTRODUCTION

Aspects of the present disclosure relate to computationally efficientmethods for forecasting values with confidence intervals based ondatasets with complex geometries (e.g., time series data).

Forecasting with simultaneous quantification of uncertainty in theforecast has emerged as a problem of practical importance for manyapplication domains, such as: computer vision, time series forecasting,natural language processing, classification, and regression, to name afew. Recent research has focused on deep learning techniques as onepossible approach to provide suitable forecasting models, and severalapproaches have been studied to characterize uncertainty in the deeplearning framework, including: dropout, Bayesian neural networks,ensemble-based models, calibration-based models, neural processes, anddeep kernel learning. Of these various approaches, deep kernel learninghas emerged as a useful framework to forecast values and characterizeuncertainty (alternatively, confidence) in the forecasted valuessimultaneously. In particular, deep kernel learning has proven usefulfor forecasting time series datasets with complex geometries.

Deep kernel learning combines deep neural network techniques withGaussian process. In this way, deep kernel learning combines thecapacity of approximating complex functions with deep neural networktechniques with the flexible uncertainty estimation framework ofGaussian process.

Unfortunately, deep kernel learning is computationallyexpensive—generally O(n³), where n is the number of training datapoints. Thus, when applied to organizations' ever larger and morecomplex datasets, deep kernel learning may require significant amountsof time and processing resources to operate. And as datasets get larger,the problem grows significantly non-linearly. Consequently,organizations are forced to invest significant resource in additionaland more powerful on-site computing resources and/or to offload theprocessing to cloud-based resources, which are expensive and which maycreate security concerns for certain types of data (e.g., financialdata, personally identifiable data, health data, etc.).

Accordingly, what is needed is a framework for reducing thecomputational complexity of deep kernel learning while still being ableto forecast and characterize uncertainty simultaneously.

BRIEF SUMMARY

Certain embodiments provide a method for performing finite rank deepkernel learning, including: receiving a training dataset; forming a setof embeddings by subjecting the training data set to a deep neuralnetwork; forming, from the set of embeddings, a plurality of dotkernels; combining the plurality of dot kernels to form a compositekernel for a Gaussian process; receiving live data from an application;and predicting a plurality of values and a plurality of uncertaintiesassociated with the plurality of values simultaneously using thecomposite kernel.

Other embodiments comprise systems configured to perform finite rankdeep kernel learning as well as non-transitory computer-readable storagemediums comprising instructions for performing finite rank deep kernellearning.

The following description and the related drawings set forth in detailcertain illustrative features of one or more embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The appended figures depict certain aspects of the one or moreembodiments and are therefore not to be considered limiting of the scopeof this disclosure.

FIG. 1 depicts an example of a hierarchy of kernel spaces.

FIG. 2 depicts a synthetic dataset that has been forecasted using afinite rank deep kernel learning method

FIG. 3 depicts finite rank orthogonal embeddings corresponding to thedataset depicted in FIG. 2.

FIGS. 4A-4D depicts a first example simulation comparing the performanceof different modeling techniques, including deep kernel learning.

FIGS. 5A-5D depicts a second example simulation comparing theperformance of different modeling techniques, including deep kernellearning.

FIG. 6 depicts a third example simulation comparing the performance ofdifferent modeling techniques, including deep kernel learning.

FIG. 7 depicts an example of a finite rank deep kernel learning flow.

FIG. 8 depicts an example of using a deep neural network to createembeddings for a composite kernel.

FIG. 9 depicts an example of a finite rank deep kernel learning method.

FIGS. 10A and 10B depict example application input and output based on afinite rank deep kernel learning model.

FIG. 11 depicts an example processing system for performing finite rankdeep kernel learning.

To facilitate understanding, identical reference numerals have beenused, where possible, to designate identical elements that are common tothe drawings. It is contemplated that elements and features of oneembodiment may be beneficially incorporated in other embodiments withoutfurther recitation.

DETAILED DESCRIPTION

Aspects of the present disclosure provide apparatuses, methods,processing systems, and computer readable mediums for computationallyefficiently forecasting values with confidence intervals, which areparticularly well-suited to operate on datasets with complex geometries.

Deep kernel learning is a state-of-the-art method to forecast withuncertainty bounds (or confidence intervals) that relies upon twounderlying machine learning paradigms, namely: deep neural networks andGaussian process. In deep kernel learning, the deep neural network isused to learn a kernel operator of a Gaussian process, which is thensuccessively used to forecast with uncertainty bounds.

Described herein is a modelling framework, which may be referred to afinite rank deep kernel learning, which beneficially reduces thecomputational complexity of deep kernel learning while enhancing deepkernel learning's ability to approximate complex functions and estimateuncertainty (or confidence). Notably, while described in the examplecontext of deep kernel learning throughout, the framework describedherein is similarly applicable to other kernel-based techniques wheredeep neural networks are used to learn the kernel, such as: deep kernellearning for classification and deep neural network-based support vectormachines, to name a few.

One feature of finite rank deep kernel learning is a composite kernel(or “expressive” kernel), which is a linear combination of a pluralityof simpler linear (or “dot”) kernels. Composite kernels are capable ofcapturing complex geometries of a dataset, such as where certain regionsof the dataset have very different structure as compared to otherregions, even verging on discontinuity. Modeling this type of dataset isdifficult with conventional machine learning algorithms, such as deepkernel learning, because traditional machine learning algorithms try tolearn a global description of a dataset.

In finite rank deep kernel learning, each dot kernel is learned by adeep neural network. Learning simpler dot kernels, which can then belinearly combined into a composite kernel, is easier for the deep neuralnetwork to learn because an individual dot kernel represents the localgeometry of the dataset rather than the global geometry, which in manycases is more complicated. Because the dot kernels are easier to learn,the overall performance of finite rank deep kernel learning is improvedfrom a processing efficiency standpoint, and any processing devicerunning finite rank deep kernel learning will enjoy improvedperformance, such as faster operation, lower processing requirements,and less memory usage, as compared to conventional machine learningmethods.

Example applications of finite rank deep kernel learning include:regression with confidence bounds, forecasting time series or sequentialdata (long and short term), and anomaly detection, as a few examples. Inregression problems based on a set of dependent variables x, the aim isto predict the value of a response variable y≈f(x)+ε. In a time seriesforecasting problem, z(t) is either a time series or a sequence. The aimis to forecast z(T+τ) based on {a,b(t),z(t)|t=0, , , , T}, where a isthe time series metadata and b(t) is the exogenous variable. The timeseries forecasting problem can be formulated as a regression problemwhere x={a,b(t),z(t)|t=0, . . . , T}, and y=z(T+τ) in the frameworkdescribed herein. In both cases, the output of the model would be aprobability distribution of the variable y.

Gaussian Process Overview

A Gaussian process is a stochastic process (e.g., a collection of randomvariables indexed by time or space) in which every finite collection ofthose random variables has a multivariate normal distribution. In otherwords, every finite linear combination of those variables is normallydistributed. The distribution of a Gaussian process is the jointdistribution of all those random variables, and as such, it is adistribution over functions with a continuous domain, e.g. time orspace.

A machine-learning algorithm that involves a Gaussian process useslearning and a measure of the similarity between points (the kernelfunction) to predict the value for an unseen point from training data.The prediction is not just an estimate for that point, but also includesuncertainty information because it is a one-dimensional Gaussiandistribution (which is the marginal distribution at that point).Gaussian process is a maximum a posteriori (MAP) framework, whichassumes a prior probability distribution over the function space of allpossible candidate regressors, and a kernel. Thus, Gaussian process is aflexible and non-parametric framework that can forecast and quantifyuncertainty simultaneously.

Generally speaking, there are two approaches to derive the theory of theGaussian process: weight space view and function space view.

In the weight space view approach, the weights of a regression areassumed to be derived from a probability distribution, which has aGaussian prior. A maximum a posteriori estimation is utilized toevaluate the posterior, which is then used to weigh predictions for testpoints for different weight configurations.

In the function space view approach, the candidate regressor functionsare assumed to be sampled from a probability distribution of functions.The kernel operator models the covariance, and the posteriordistribution is used to average the predictions made by individualregressors. The following outlines the equations for the function spaceviewpoint.

Initially, let X={x₁, . . . x_(n)} be the training features, wherex_(i)∈

^(d), and f(X):={f(x₁), . . . f(x_(n))} is sampled from a distribution

(0, K_(X,X)), where K_(X,X)∈

^(n×n) is comprised of the value of the kernel operator evaluated atevery pair of training data points. Further, let y denote the responsevariable corresponding to the training data points. It can be shownthat:f _(*) |X _(*) ,X,y,γ,σ ²˜

(E[f _(*)], cov[f _(*)]),E[f _(*)]=K _(X) _(*) _(,X)(K _(X,X)+σ² I)⁻¹ y,Cov[f _(*)]=K _(X) _(*) _(,X) _(*) −K _(X) _(*) _(,X)[K _(X,X)+σ² I]⁻¹y.

Selection of the kernel function in Gaussian process is a non-trivialtask, as depending on the geometry of the data, the right similarity orkernel function needs to be identified.

A loss function may be constructed as the log likelihood on theposterior distribution of the Gaussian process, and during the modeltraining, the error is successively back-propagated to find the optimalembedding and the radial basis function scale parameter.

Notably, the algorithmic complexity of Gaussian process is O(n³), wheren is the number of training data points. In order to reduce thecomputational complexity, a framework is described herein to derive thekernel with a complexity of O(n²) without any approximation of thekernel.

Deep Kernel Learning Overview

In the deep kernel learning framework, a deep neural network is used tolearn an embedding, which is acted upon by a radial basis function (RBF)to create the kernel. A radial basis function (RBF) is a real-valuedfunction ϕ whose value depends only on the distance from the origin, sothat ϕ(x)=ϕ(∥x∥); or alternatively on the distance from some other pointc, called a center, so that ϕ(x,c)=ϕ(∥x−c∥). Sums of radial basisfunctions may be used to approximate given functions. This approximationprocess can also be interpreted as a simple kind of neural network.

The imposition of a radial basis function kernel adds an additionalstructure to the kernel operator, which is a continuous operator.However, such kernels may not be adequate to represent the geometry ofan arbitrary dataset, especially in cases where the dataset has widelyvarying local geometries. Thus, as described further herein, the kernelmay be created instead as a linear combination of a set of simpler dotkernels. The decomposition of the kernel in this manner (i.e., using aplurality of dot kernels) ensures that each kernel captures the localgeometry of a portion of the dataset and the linearly combined(composite) kernel captures the combined geometry of the whole dataset.As described further below, this may be achieved by constructingorthogonal embeddings as deep neural network outputs. This approachallows simultaneous unsupervised learning (e.g., clustering) andsupervised learning (e.g., regression) in a unified framework.

The Gaussian process kernel may be defined as k(x_(i),x_(j)) and akernel of deep kernel learning may be defined as:k(x _(i) ,x _(j))→k(ϕ(x _(i),ω),ϕ(x _(j),ω)|ω,θ)where, ϕ(x_(i),ω) is the embedding learnt by the deep neural network.The set of parameters are defined as γ=[ω,θ]. The log marginallikelihood function

is minimized for the Gaussian process, and back-propagation is used tooptimize simultaneously ω and θ, which gives the following equations:

${\frac{\partial\mathcal{L}}{\partial\theta} = {\frac{\partial\mathcal{L}}{\partial K_{\gamma}}\frac{\partial K_{\gamma}}{\partial\theta}}},{\frac{\partial\mathcal{L}}{\partial\omega} = {\frac{\partial\mathcal{L}}{\partial K_{\gamma}}\frac{\partial K_{\gamma}}{\partial{\phi\left( {x_{i},\omega} \right)}}\frac{\partial{\phi\left( {x_{i},\omega} \right)}}{\partial\omega}}},{\frac{\partial\mathcal{L}}{\partial K_{\gamma}} = {\frac{1}{2}{\left( {{K_{\gamma}^{- 1}{yy}^{T}K_{\gamma}^{- 1}} - K_{\gamma}^{- 1}} \right).}}}$

Notably, the complexity of computing K_(γ) ⁻¹ is

(n³). By contrast, in finite rank deep kernel learning, a deep neuralnetwork is configured to learn the embeddings as mutually orthogonal,which is successively used to reduce the computational complexity incomputation of K_(γ) ⁻¹ to

(n²).

Two challenges of conventional deep kernel learning are reducingcomputational cost without approximating the kernel function and therepresentation power of the kernel function. The finite rank deep kernellearning framework described herein enhances the representation power,while at the same time reducing the computational complexity. In otherwords, the framework described herein improves the performance of anymachine upon which it is running (through reduced computationcomplexity) as well as improves the performance of whatever applicationit is supporting (through improved representation power).

Hierarchy of Kernel Operators

Let X be an arbitrary topological space, which is a feature space of aregression. Let H be the Hilbert space of the bounded real valuedfunctions defined on X. Initially, a kernel operator K:X×X→

is called positive definite when the following is true:K(x,y)=K(y,x)Σ_(i,j=1) ^(n) c _(i) c _(j) K(x _(i) ,x _(j))≥0,∀n∈

,x _(i) ,x _(j) ∈X,c _(i) ∈R.

With the aid of the Riesz Representation Theorem, it can be shown thatfor all x∈X, there exists an element K_(x)∈H, such that f(x)=

f,L_(x)

, where

·,·

is an inner product, with which the Hilbert Space H is endowed. Next, areproducing kernel for the Hilbert space H (RKHS) may be defined, whichconstructs an operator K(x,y) as an inner product of two elements K_(x)and K_(y) from the Hilbert space H. A reproducing kernel for the Hilbertspace H may be defined as:K(x,y):=

K _(x) ,K _(y)

,∀x,y∈X.

From the definition of the reproducing kernel for the Hilbert space H,it can be observed that the reproducing kernel for the Hilbert space Hsatisfies the conditions of the positive definite kernel operators, asdescribed above. Moreover, the Moore Aronszajn Theorem proves that forany symmetric positive definite kernel operator K, there exists aHilbert space H for which it is the reproducing kernel for the Hilbertspace H, or in other words, the operator satisfies K(x,y):=

K_(x),K_(y)

, where K_(x) and K_(y) belongs to the Hilbert space H of the realvalued bounded functions on X. Notably, K_(x) and K_(y) can bediscontinuous.

It also can be noted that the space of the reproducing kernel for theHilbert space H may be very rich in terms of the complexity, and no apriori assumption need be made on the smoothness of the operator. Anexample of a reproducing kernel for the Hilbert space H, which isnon-smooth is as follows:

δ(x, y) = 1 , if  x = y,  = 0, otherwise.

Thus, K=δ is a symmetric and positive definite kernel, but isnon-smooth.

Next, a subclass of the reproducing kernel for the Hilbert space H maybe considered, which is continuous in addition to being symmetric andpositive definite. Such kernels are called Mercer kernels. Mercer'sDecomposition Theorem provides a decomposition of such an arbitrarykernel into the Eigen functions, which are continuous themselves. Forexample, for any continuous reproducing kernel for the Hilbert spaceK(x,y), the following condition is satisfied:

${{\lim\limits_{R\rightarrow\infty}{\sup\limits_{x,y}{{{K\left( {x,y} \right)} - {\sum\limits_{i = 1}^{R}{\zeta_{i}{\Theta_{i}(x)}{\Theta_{i}(y)}}}}}}} = 0},$

where Θ_(i)∈

⁰ forms a set of orthonormal bases, and ζ_(i)∈

⁺ are the i^(th) Eigen function and Eigen value of the integral operatorT_(k)(⋅), corresponding to the kernel K. It also can be shown with theaid of the spectral theorem that the Eigen values asymptoticallyconverge to 0.

Kernel operators may be thought of as similarity functions that capturerelationships between points in a dataset. FIG. 1 depicts an example ofa hierarchy of kernel spaces, including rank 1 kernels 102, finite rankkernels 104, Mercer kernels 106, and reproducing kernels for the Hilbertspace 108.

Kernel functions used in existing deep kernel learning methods areprimarily radial basis function kernels and polynomial kernels andconsequently form a small set of possible kernel functions to representa potentially rich and complex data set. These kernels are primarilyrank 1 (e.g., 102), which constitute a smaller subspace of possiblekernels as depicted in FIG. 1.

By contrast, finite rank deep kernel learning expresses a compositekernel as a sum of multiple simpler dot kernels, which cover the spaceof finite rank Mercer kernels 104, as depicted in FIG. 1. The compositekernels may be expressed as follows:K(x,y)=Σ_(i=1) ^(R)ϕ_(i)(x)·ϕ_(i)(y),where ϕ_(i)(x)'s form a set of orthogonal embeddings, which are learntby a deep neural network. By expressing the composite kernel in thisfashion, one can show that the possible set of kernels would becomericher than the existing kernels adopted in conventional deep kernellearning approaches.

As depicted in FIG. 1, Mercer kernels 106 form a smaller subspace of thereproducing kernels for the Hilbert space 108, as Mercer kernels aregenerally continuous. For instance, the kernel δ(x,y), which is areproducing kernel for the Hilbert space, but is not continuous, cannotbe decomposed according to the Mercer's Decomposition Theorem. But arich subset of kernels can be represented by Mercer kernels, which canbe expressed as follows:K(x,y)˜Σ_(i=1) ^(R)ζ_(i)Θ_(i)(x)Θ_(i)(y),

where Θ_(i) forms an orthonormal basis. The orthonormality of the basisensures an inverse of the operator can be constructed as follows:

${\sum\limits_{i = 1}^{R}{\frac{1}{\zeta_{i}}{\Theta_{i}(x)}{\Theta_{i}(y)}}},$

which reduces computation of the inverse operator. Further, the Mercerkernels 106 can have countable ranks with diminishing Eigen values. So,while Mercer kernels 106 are a less rich set as compared to thereproducing kernels for the Hilbert space 108, they nevertheless form adiverse subspace of the reproducing kernels for the Hilbert space 108.

Notably, a Mercer kernel generally has countable rank i.e. any Mercerkernel can be expressed as sum of countable rank 1 kernels. For example,Θ_(i)(x)Θ_(i)(y) is a rank 1 kernel. Generally, a Mercer kernel K(x,y)is of finite rank R if it can be expressed as follows:K(x,y)=Σ_(i=1) ^(R)ζ_(j)Θ_(i)(x)Θ_(i)(y).

In some cases, kernels used in machine learning are rank 1 Mercerkernels, which have σ₁=1, and σ_(i)=0 for i≥2. For example, popularkernels used in machine learning, such as a polynomial kernel(k(x,y)=(x′y+c)^(d)) and a Radial Basis Function kernel

$\left( {{k\left( {x,y} \right)} = {\exp\left( {- \frac{{{x - y}}^{2}}{\sigma^{2}}} \right)}} \right),$are rank 1 Mercer kernels. Generally, any rank 1 Mercer kernel may beexpressed as K(x,y)=

c(x),c(y)

for some continuous function c.

As above, Mercer kernels 106 form a subspace of the reproducing kernelsfor the Hilbert space 108. Similarly, rank 1 Mercer kernels 102 form asmaller subspace of the Mercer kernels 106, as depicted in FIG. 1.

Finite Rank Deep Kernel Learning

It is desirable to create a set of kernels that have greaterrepresentational power. One method is to use a finite rank Mercer kernelto represent a richer class of kernels, which may be represented asfollows:K(x,y)=Σ_(i=1) ^(R)ϕ_(i)(x,ω)ϕ_(i)(y,ω)

This kernel selection technique is useful when using deep neuralnetworks to learn the embeddings ϕ_(i), especially where a dataset haswidely differing local geometries, because the deep neural network coulddecompose the embeddings ϕ_(i) into orthogonal embeddings.

Notably, any arbitrary finite rank Mercer kernel can be approximated bya deep neural network. Thus, for any Mercer kernel:K(x,y)=Σ_(i=1) ^(R)ζ_(i)Θ_(i)(x)Θ_(i)(y),and an ε>0,

there exists an N and a family of neural network regressors with finitenumber of hidden units and output layer ϕ_(i)(z,w) such that:

${{{{K\left( {x,y} \right)} - {\sum\limits_{i = 1}^{R}{{\phi_{i}\left( {x,w} \right)}{\phi_{i}\left( {y,w} \right)}}}}}\left\langle {ɛ,\mspace{11mu}{\forall R}} \right\rangle\mspace{14mu} N},{{and}\mspace{14mu}{\forall x}},{y \in X},$

where Θ_(i)(z,w) forms a set of orthogonal functions. Accordingly, anarbitrary smooth Mercer kernel can be modeled by a multi-layer neuralnetwork with outputs Θ_(i)(z,w). The outputs of the neural network mayform embeddings that are orthogonal to one another. As a consequence ofthe orthogonality, the inverse operator can also be expressed in termsof the deep neural network output layer.

Generation of Orthogonal Embeddings

The Gaussian process kernel may be modeled as follows:K(x,y)=Σ_(i=1) ^(R)ϕ_(i)(x,w)ϕ_(i)(y,w),where ϕ_(i)(y,w) would be ideally orthogonal to one another. The deepkernel learning algorithm optimizes the negative log likelihoodfunction, based on the kernel operator, which may be represented asfollows:−log p(y|x)˜y ^(T)(K _(γ)+σ² I)⁻¹ y+log|K _(γ)+σ² I|

A penalty term to the cost may be introduced, as follows:

${{- \log}\;{p\left( {y❘x} \right)}} + {\lambda{\sum\limits_{i,j,{i \neq j}}\left( {{\phi_{i}\left( {x,w} \right)}^{T}{\phi_{j}\left( {x,w} \right)}} \right)^{2}}}$where, λ is a weight assigned to the orthogonality objective as opposedto the log likelihood. Notably, Σ_(i,j,i≠j)(ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))²is minimized when the embeddings are orthogonal.

Below it will be shown further that the inversion of the matrix(K_(γ)+σ²I)⁻¹, and the determinant computation |K_(γ)+σ²I| can befurther simplified, and as a consequence the optimization can be done inbatches.

Computational Complexity

The main computational bottleneck in deep kernel learning concerns theinversion of the kernel operator, which as above has computationalcomplexity O(n³). By contrast, expressing the Gaussian process kernel asa linear combination of dot kernels as with finite rank deep kernellearning means that the composite kernel operator can be inverted asfollows:

${K\left( {x,y} \right)} = {\sum\limits_{i = 1}^{R}{{\frac{1}{{\phi_{i}(x)}} \cdot \frac{1}{{\phi_{j}(x)}}}{{\phi_{i}(x)} \cdot {\phi_{i}(y)}}}}$

This construction reduces the complexity to O(n²) i.e., an order ofmagnitude. This reduction in complexity significantly improves theperformance of finite rank deep kernel learning over conventional deepkernel learning. For example, processing systems implementing finiterank deep kernel learning will have to expend fewer processing resourcesand memory resources, and the processing itself will completesignificantly quicker, thereby improving the performance of theprocessing machine as well as the application being supported by theprocessing machine.

The following equations demonstrate the reduction in computationalcomplexity in more detail. As above, the kernel operator may beexpressed as follows:K(x,y)=Σ_(i=1) ^(R)ϕ_(i)(x,w)ϕ_(i)(y,w),

where the embeddings ϕ_(i)(y,w) are orthogonal to each other. For afinite number of training data points, the embeddings may be defined asfollows:Φ_(i)=[ϕ_(i)((x ₁ ,w)), . . . ϕ_(i)((x _(n) ,w))]

The kernel matrix is thus defined as:K=Σ _(i=1) ^(R)Φ_(i)·Φ_(i) ^(T).

The orthogonality is also useful in terms of reducing the computationalcomplexity, while inverting the kernel operator (as discussed above):

${K^{- 1}\left( {x,y} \right)} = {\sum\limits_{i = 1}^{R}{{\frac{1}{\Phi_{i}} \cdot \Phi_{i}}\Phi_{i}^{T}}}$

It can be shown using the orthogonality property of the embeddings that:

${{K\left( {x,y} \right)} \cdot {K^{- 1}\left( {x,y} \right)}} = {{\left( {\sum\limits_{i = 1}^{R}{\Phi_{i} \cdot \Phi_{i}^{T}}} \right) \cdot \left( {\sum\limits_{i = 1}^{R}{{\frac{1}{{\Phi_{i}}^{2}} \cdot \Phi_{i}}\Phi_{i}^{T}}} \right)} = {I.}}$

Next, an expression may be derived for (K+σ²I)⁻¹ from the kerneloperator. In particular, an identity matrix may be expressed as:

$I = \left( {{\sum\limits_{i = 1}^{R}{{\frac{1}{{\Phi_{i}}^{2}} \cdot \Phi_{i}}\Phi_{i}^{T}}} + {\sum\limits_{j = 1}^{n - R}{{\frac{1}{{\eta_{j}}^{2}} \cdot \eta_{j}}\eta_{j}^{T}}}} \right)$

where η_(j)'s are orthogonal to each other and form the null space ofthe subspace spanned by the embeddings Φ_(i). This construction leads tothe following equations, which provides the formulae to invert matrices,which may be used for training and inference with reduced computationalcomplexity:

$\mspace{20mu}{\left( {K + {\sigma^{2}I}} \right) = {{\sum\limits_{i = 1}^{R}{{\frac{\sigma^{2}}{{\Phi_{i}}^{2}} \cdot \Phi_{i}}\Phi_{i}^{T}}} + {\sigma^{2}{\sum\limits_{j = 1}^{n - R}{{\frac{1}{{\eta_{j}}^{2}} \cdot \eta_{j}}\eta_{j}^{T}}}}}}$${K_{*}\left( {K + {\sigma^{2}I}} \right)}^{- 1} = {{{\sum\limits_{i = 1}^{R}{{\frac{\sigma^{2}}{{\Phi_{i}}^{2}} \cdot \Phi_{i}}\Phi_{i}^{T}}} + {\sum\limits_{j = 1}^{n - R}{{\frac{\sigma^{2}}{{\eta_{j}}^{2}} \cdot \eta_{j}}\eta_{j}^{T}}}} = {\left( {\sum\limits_{i = 1}^{R}{\Phi_{i}^{*} \cdot \left( \Phi_{i}^{*} \right)^{T}}} \right) \cdot {\left( {\sum\limits_{i = 1}^{R}{{\frac{\sigma^{2}}{{\Phi_{i}}^{2}} \cdot \Phi_{i}}\Phi_{i}^{T}}} \right).}}}$

The next step addresses the question of whether or not the loss functioncan be decomposed over the training data points. The following set ofequations leads to the decomposition of the cost function into trainingdata points:

${{{- \log}\;{p\left( {y❘x} \right)}\text{∼}{y^{T}\left( {K_{\gamma} + {\sigma^{2}I}} \right)}^{- 1}y} + {\log{{K_{\gamma} + {\sigma^{2}I}}}}} = {{{{y^{T}\left( {{\sum\limits_{i = 1}^{R}{\Phi_{i}\Phi_{i}^{T}}} + {\sigma^{2}I}} \right)}^{- 1}y} + {\log{{{\sum\limits_{i = 1}^{R}{\Phi_{i}\Phi_{i}^{T}}} + {\sigma^{2}I}}}}} = {{{{y^{T}\left( {\prod\limits_{i = 1}^{R}\left( {{\frac{1}{\sigma^{2}}\Phi_{i}\Phi_{i}^{T}} + {\sigma^{2}I}} \right)} \right)}^{- 1}y} + {\log{{\prod\limits_{i = 1}^{R}\left( {{\frac{1}{\sigma^{2}}\Phi_{i}\Phi_{i}^{T}} + {\sigma^{2}I}} \right)}}}} = {{{y^{T}\left( {\prod\limits_{i = 1}^{R}\left( {{\frac{1}{\sigma^{2}}\Phi_{i}\Phi_{i}^{T}} + {\sigma^{2}I}} \right)^{- 1}} \right)}y} + {\log{{\prod\limits_{i = 1}^{R}\left( {{\frac{1}{\sigma^{2}}\Phi_{i}\Phi_{i}^{T}} + {\sigma^{2}I}} \right)}}}}}}$

Notably, the preceding formulas are derived assuming orthogonality ofthe embeddings ϕ_(i). However, they would be approximately orthogonalsince orthogonality has been modeled as a continuous loss function.

Deep Orthogonal Bayesian Linear Regression

Parallels may be drawn between the Gaussian process and Bayesian linearregression. In fact, Bayesian linear regression can be shown asequivalent to Gaussian process when employing dot kernels, as describedabove. And similar to Gaussian process, a set of orthogonal embeddingfunctions can be used in Bayesian linear regression as follows:

${{y\text{∼}{\sum\limits_{i = 1}^{R}{{\phi_{i}(x)}w_{i}}}} + ɛ},$

where, ε is the observation noise with variance σ², and R is the rank ofthe combined embedding. Therefore, it can be shown that:

${f_{*}❘x_{*}},X,{y\text{∼}{\mathcal{N}\left( {{\frac{1}{\sigma^{2}}{\sum\limits_{i = 1}^{R}{{\phi_{i}\left( x_{*} \right)}A^{- 1}\Phi_{i}y}}},{{\phi_{i}\left( x_{*} \right)}A^{- 1}{\phi\left( x_{*} \right)}}} \right)}},$

where Φ_(i)=Φ_(i)(X), and

$A = {{\sum\limits_{i = 1}^{R}{\frac{1}{\sigma^{2}}\Phi_{i}\Phi_{i}^{T}}} + {\sum\limits_{p}^{- 1}.}}$This can be further modified as follows:

${f_{*}❘x_{*}},X,{y\text{∼}{\mathcal{N}\left( {{\sum\limits_{i = 1}^{R}{{\phi_{i}\left( x_{*} \right)}{\sum_{p}{{\Phi_{i}\left( {K + {\sigma^{2}I}} \right)}^{- 1}y}}}},{\sum\limits_{i = 1}^{R}\left( {{{\phi_{i}\left( x_{*} \right)}{\sum_{p}{\phi\left( x_{*} \right)}}} - {{\phi_{i}\left( x_{*} \right)}{\sum_{p}{{\Phi_{i}\left( {K + {\sigma^{2}I}} \right)}^{- 1}\Phi_{i}^{T}{\sum_{p}{\phi_{i}\left( x_{*} \right)}}}}}} \right)}} \right)}},$

where K=Σ_(i=1) ^(R) Φ_(i)Σ_(p)Φ_(i). With orthogonality of thepreceding formula, the complexity of (K+σ²I)⁻¹ may be reduced asdemonstrated above. Thus, a global optimization problem may beformulated, which would optimize the log likelihood of the Bayesianlinear regression with orthogonal embeddings, learnt by a deep neuralnetwork. And the orthogonality constraint in the loss function may beimplemented as described above.

Example Simulations of Finite Rank Deep Kernel Learning

FIG. 2 depicts a synthetic dataset that has been forecasted using afinite rank deep kernel learning method as described herein. Thecorresponding finite rank orthogonal embeddings ϕ_(i)(x) (based on thedataset depicted in FIG. 2) are depicted in FIG. 3. As depicted, thedataset in FIG. 2 has a complex geometry wherein different portions 202,204, and 206 of the dataset have distinctly different geometries.Depicted in FIG. 2 are training samples, a “true” function, a predictedfunction (via finite rank deep kernel learning), and a predictedstandard deviation, which acts as an uncertainty or confidence interval.

As depicted in FIG. 2, the predicted function closely approximates thetrue function in each of the different portions (202, 204, and 206),despite the very different geometries in those sections.

FIG. 3 depicts an example of the approximately orthogonal embeddingsϕ_(i)(x) (302) that form the basis of the dot kernels, which whencombined into a composite (i.e., expressive) kernel, create the outputdepicted in FIG. 2. The orthogonality of the embeddings shows theexpressive power of the linear combination of dot kernels (i.e.,composite kernel). In particular, these dot kernels tease out the localrelationships in the underlying dataset. The dot kernels can further becombined to form the Gaussian process kernel. Hence, the dot kernelsinherently identify clusters of data in the dataset while learning thepattern of the overall dataset simultaneously. The clustering of thedataset into groups allows the neural network to learn the localgeometries in a decoupled fashion. The benefit of this approach is thatthe deep neural network can more accurately fit regions of the datasetin cases of datasets with discontinuities.

FIGS. 4A-4D depicts a first example simulation comparing the performanceof different modeling techniques, including deep kernel learningdepicted in FIG. 4A, Gaussian process depicted in FIG. 4B, bag of neuralnetworks depicted in FIG. 4C, and finite rank deep kernel learningdepicted in FIG. 4D, which is described herein, using a time seriesdataset. In this example, the time series dataset is based on asinusoidal function whose frequency increases as the square of x.Additionally, heteroscedastic noise has been added to the function suchthat the noise magnitude increases from the left to right.

It is apparent from the simulation results that the bag of neuralnetwork method of FIG. 4C underestimates the confidence intervals nearthe noisy region and overestimates the confidence in the high frequencyregion. For both deep kernel learning of FIG. 4A and Gaussian process ofFIG. 4B, it is apparent that the confidence intervals fluctuate heavilynear the noisy region.

By contrast, the finite rank deep kernel learning method of FIG. 4Dproduces confidence bounds that are relatively stable and that captureregions of high noise and fluctuations. Thus, the finite rank deepkernel learning method shows significant improvement in the ability toboth forecast values based on a data set with complex geometry whileproviding accurate quantification of uncertainty via confidenceintervals as compared to the conventional methods depicted in FIGS.4A-4C.

FIGS. 5A-5D depicts simulation results for a logistic map example. Inthis example, the logistic map is a chaotic but deterministic dynamicalsystem x_(n+1)=rx_(n)(1−x_(n)), where x_(n)∈S₁. The time series data isgenerated by the system for r=4.1, which falls in the region of strangeattractor. In general, strange attractor signifies deterministic chaos,which is difficult to forecast. Thus, performance of a modelingframework, such as finite rank deep kernel learning, may be assessed by,for example, modeling a time series that is deterministically chaotic.

In this example, the Gaussian process of FIG. 5B and bag of neuralnetwork of FIG. 5C have outputs with overly wide confidence intervals,and these models erroneously identify chaos as noise. The deep kernellearning output of FIG. 5A has confidence bounds that are relativelymoderate, but does not track the true function particularly closely.

By contrast, the finite rank deep kernel learning method of FIG. 5Dcorrectly captures the chaotic time series with very narrow confidencebounds.

FIG. 6 is an example of simulation results based on a regressiondataset. In this case, a normalized root mean squared error wascalculated as a measure of accuracy, which is computed as the root meansquared error of a predictor divided by the standard error of thesamples. In general, a normalized root mean squared error<1 would be athreshold for any predictor performing better than the sample mean.

In this example, the normalized root mean squared error values werefound to be 0.41 for deep kernel learning (a) and 0.25 for finite rankdeep kernel learning (d)—representing a near 40% improvement in modelpredictive performance. Further, the average CPU time lapsed for oneepoch during the model training was 0.32 sec for deep kernel learningand 0.079 sec for finite rank deep kernel learning—representing a near76% improvement. Further, the inference time was 0.03 seconds for deepkernel learning and 0.01 seconds for finite rank deep kernellearning—representing a 67% performance improvement.

Taken collectively, FIGS. 4-6 demonstrate that finite rank deep kernellearning outperforms conventional modeling methods, including deepkernel learning, both in terms of accuracy as well as computationalefficiency.

Example Finite Rank Deep Kernel Learning Flow

FIG. 7. depicts a finite rank deep kernel learning flow 700. Flow 700begins at step 702 with acquiring training data. In some cases, asdescribed above, the training data may form a complex geometry.

Flow 700 then proceeds to step 704 where the training data is fed to adeep neural network. The outputs of the deep neural network areembeddings. As described above, by changing the loss function toemphasize orthogonality of the embeddings, the resulting embeddings maybe approximately orthogonal. FIG. 8, described below, depicts an exampleof using a deep neural network to construct embeddings.

Flow 700 then proceeds to step 706 where a plurality of dot kernels areconstructed from the approximately orthogonal embeddings. As describedabove, each of the dot kernels may be a finite rank Mercer kernel. FIG.8, described below, depicts an example of forming a dot kernel fromembeddings produced by a deep neural network.

Flow 700 then proceeds to step 708 where the plurality of dot kernelsare linearly combined into a composite (i.e., expressive) kernel.

Flow 700 then proceeds to step 710 where the composite kernel is used asthe basis of a finite rank deep kernel learning predictive model. Forexample, the predictive model may act on live data 712 to createpredictions. Notably, the predictive model has two separate outputs,including the predicted values, which are mean values based on thepredictions from each dot kernel, and a confidence associated with thepredicted values.

Flow 700 concludes at step 714 where an application, such as computervision, time series forecasting, natural language processing,classification, and regression, or those described in more detail below,uses the outputs of the predictive model to provide anapplication-specific output 716.

Example of Using Deep Neural Network to Create Embeddings for aComposite Kernel

FIG. 8 depicts an example of using a deep neural network to createembeddings for a composite kernel for finite rank deep kernel learning.As depicted in FIG. 8, a deep neural network with multiple hidden layers802 learns a plurality of embeddings 804, which in this example includeϕ₁(x), ϕ₂(x), and ϕ₃(x). The plurality of embeddings are used to formthe dot kernel for the Gaussian process 806, which in this example isrank 3. In this example, the deep neural network produces vector output,each entry of which forms an individual embedding.

Example Method for Creating Finite Rank Deep Kernel Learning Model

FIG. 9 depicts a finite rank deep kernel learning method 900.

Method 900 begins at step 902 with receiving a training dataset. Asdescribed above, in some examples the training dataset may comprisetraining data with complex geometries, such as datasets includingmultiple discontinuous portions.

Method 900 then proceeds to step 904 with forming a set of embeddings bysubjecting the training data set to a deep neural network.

Method 900 then proceeds to step 906 with forming, from the set ofembeddings, a plurality of dot kernels. In some examples, Σ_(i,j,i≠j)(ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))² is minimized as a cost function to maintainan orthogonality of the set of embeddings when forming the set ofembeddings.

In some examples, the orthogonality of the set of embeddings isoptimized based on a cost function, wherein the cost function includes apenalty term λ associated with orthogonality of the set of embeddings.For example, the cost function may be implemented as: −logp(y|x)+λΣ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))².

Method 900 then proceeds to step 908 with combining the plurality of dotkernels to form a composite kernel. In one example, the composite Kernelfor the Gaussian process is modeled as a linear combination of theplurality of dot kernels as: K(x,y)=Σ_(i=1) ^(R) ϕ_(i)(y,w). In someexamples, the composite kernel for the Gaussian process is a finite rankMercer kernel.

Method 900 then proceeds to step 910 with receiving live data from anapplication. In this example, live data is distinguished from trainingdata in that it is data the model has not yet seen. In one example, thelive data comprises financial data. In another example, the live datacomprises resource utilization data. In yet another example, the livedata is user activity data (e.g., user log data captured on systems usedby users or accessed by users).

Method 900 then proceeds to step 912 with predicting a plurality ofvalues and a plurality of uncertainties associated with the plurality ofvalues simultaneously using the composite kernel.

In one example, the application is a financial management application,the plurality of values comprises a plurality of predicted futurefinancial transactions, and each uncertainty of the plurality ofuncertainties associated with a respective predicted future financialtransaction estimates a range of values of the respective predictedfuture transaction.

In another example, the application is a resource managementapplication, the plurality of values comprises a plurality of predictedresources needs, and each uncertainty of the plurality of uncertaintiesassociated with a respective predicted future resource need estimates arange of values of the respective resource need.

In yet another example, the application is a resource access controlapplication, the plurality of values comprises a plurality of predicteduser activities, and each uncertainty of the plurality of uncertaintiesassociated with a respective predicted future user activity estimates arange of values of the respective user activity.

Notably, FIG. 9 is just one example, and other methods includingdifferent aspects are described herein.

Example Applications for Finite Rank Deep Kernel Learning

Finite rank deep kernel learning is broadly applicable to anyapplication where forecasting or prediction with uncertainty estimation(e.g., confidence intervals) is necessary. In particular, finite rankdeep kernel learning may improve application and processing systemperformance where the underlying dataset is large and complex (e.g., interms of geometry).

A first example application is cash flow forecasting e.g., for providingfinancial services. For example, a financial services organization mayseek to provide cash flow forecasting services to users of a financialplanning application. Because cash flow, when considered as a timeseries dataset, tends to be complex and discontinuous, finite rank deepkernel learning (as described herein) is a desirable methodology.Moreover, because the organization offering the financial services mayhave many customers with much data that can be aggregated, finite rankdeep kernel learning may significantly improve the performance of thefinancial planning application in terms of accuracy of cash flowforecasts, confidence in cash flow forecasts, speed of generating theforecasts, and efficiency of computing resources used to produce theforecasts and uncertainty estimations (e.g., confidence intervals).

FIGS. 10A and 10B depict example application input and output based on afinite rank deep kernel learning model. As described above, a finiterank deep kernel learning model may be used to forecast financialtransactions, such as a user's financial transactions over time,concurrently with a prediction of the confidence of those forecasts.

FIG. 10A depicts a table of historical user financial transaction data1002, which includes data related to multiple users (e.g., U1-U3), arange of dates of transactions, and transaction amounts. The data intable 1002 may be processed by a finite rank deep kernel leaning modelto produce output including forecasted values as well as an uncertaintymeasure for the forecasted values as depicted in table 1004. Theuncertainty measure may be used to form a confidence interval, asdepicted in FIG. 10B, which depicts forecasted data for one of the usersin tables 1002 and 1004. In this example, confidence interval 1010 is a95% confidence interval, but in other examples, it may be any otherconfidence interval.

FIG. 10B depicts a graphical forecast output showing actual data 1006 aswell as the forecasted data 1008 with a confidence interval 1010. Asseen here again, despite having an irregular geometry, the finite rankdeep kernel learning model is able to accurately model the actual datawhile providing simultaneous and time-dependent uncertaintyquantification.

The prediction of financial transactions (e.g., cash flow) over timealong with confidence intervals can improve the types of financialservices offered to users. This is especially true where an organizationwants to limit “false positive” alerts, such as an alert that the user'spredicted cash flow would underrun their current bank account balance.Thus, the confidence interval can improve the ability to tune suchalerts for user experience.

A second example application is forecasting computer resourceutilization, including local and cloud-based resources. Computerresource utilization is particularly challenging to forecast over longerperiods of time given it has cyclical and non-cyclical elements inaddition to significant variability at any given time. This sort ofcomplex geometry time series data is again an excellent candidate foranalysis by a finite rank deep kernel learning model. In particular, inaddition to creating better forecasts of resource utilization (andtherefore needs) as compared to conventional modelling methods, finiterank deep kernel learning provides more accurate confidence intervals onthe forecasts, which allows for strategic planning. For example,accurately forecasting computing resource utilization may create theopportunity to contract for cloud-based resources well in advance forbetter prices than spot prices for on-demand needs. Similarly,accurately forecasting computing resource utilization may create theopportunity to plan for resource expansion and allocate budget over alonger period of time. Further yet, the use of finite rank deep kernellearning may be a contributing factor in reducing resource utilizationbased on its significantly more efficient performance characteristics ascompared to conventional modelling methods.

A third example application is detecting anomalous behavior to identifysecurity risks e.g., from user log data or cloud based application logs.By nature, the more data captured regarding user behavior provides formore opportunities for detecting anomalies. But historically,significantly increasing the amount of captured user data (e.g., logdata), meant likewise significantly increasing the amount of data thatneeded processing, the time to process it, the time to train models,etc. However, by using a computationally efficient method such as finiterank deep kernel learning, more data can be captured and processedwithout the conventional unsustainable increase in processing demand.Further, finite rank deep kernel learning creates more accurateconfidence bounds, the detection accuracy (e.g., of anomalous behavior)is improved because the confidence of what is and is not anomalous islikewise improved.

A fourth example application is resource planning, for example, foronline customer support. Some organizations may have significantlyvariable seasonal human resource needs. By way of example, a taxpreparation organization may require significantly more human resourcesduring tax preparation season as compared to the off-season. However,this sort of need is difficult to forecast given the complex nature ofthe underlying data. Finite rank deep kernel learning is well-suited forthis task because it can learn with significant granularity localpatterns in data sets. For example, as described above, the dot kernelsprovide a means for capturing localized data trends in a dataset whilestill creating a forecast that matches the characteristics of thedataset as a whole.

Many other example applications exist. Because the aforementionedapplication examples all likely involve very large data sets, the finiterank deep kernel learning method disclosed herein would significantimprove processing system performance in terms of processing cycles,total cycle time, memory usage, and others.

Example Processing System

FIG. 11 depicts an example processing system 1100 for performing finiterank deep kernel learning. For example, processing system 1100 may beconfigured to perform one or more aspects of flow 700 described withrespect to FIG. 7 and method 900 described with respect to FIG. 9.

Processing system 1100 includes a CPU 1102 connected to a data bus 1130.CPU 1102 is configured to process computer-executable instructions,e.g., stored in memory 1110 or storage 1120, and to cause processingsystem 1100 to perform methods as described herein, for example withrespect to FIG. 9. CPU 1102 is included to be representative of a singleCPU, multiple CPUs, a single CPU having multiple processing cores, andother forms of processing architecture capable of executingcomputer-executable instructions.

Processing system 1100 further includes input/output device(s) 1104 andinput/output interface(s) 1106, which allow processing system 1100 tointerface with input/output devices, such as, for example, keyboards,displays, mouse devices, pen input, and other devices that allow forinteraction with processing system 1100.

Processing system 1100 further includes network interface 1108, whichprovides processing system 1100 with access to external networks, suchas network 1114.

Processing system 1100 further includes memory 1110, which in thisexample includes a plurality of components.

For example, memory 1110 includes deep neural network component 1112,which is configured to perform deep neural network functions asdescribed above.

Memory 1110 further includes embedding component 1114, which isconfigured to determine embeddings based on output from neural deepneural network component 1112. For example, embedding component 1114 mayidentify orthogonal embeddings for use in creating a plurality of dotkernels.

Memory 1110 further includes dot kernel component 1116, which isconfigured to determine dot kernels based on the embeddings determinedby embedding component 1114.

Memory 1110 further includes composite kernel component 1118, which isconfigured to create composite (i.e., expressive) kernels from the dotkernels determined by dot kernel component 1116. For example, compositekernel component 1118 may be configured to linearly combine a pluralityof dot kernels to determine a composite kernel.

Note that while shown as a single memory 1110 in FIG. 11 for simplicity,the various aspects stored in memory 1110 may be stored in differentphysical memories, but all accessible CPU 1102 via internal dataconnections, such as bus 1130.

Processing system 1100 further includes storage 1120, which in thisexample includes training data 1122, live data 1124, and modelparameters 1126. Training data 1122 may be, as described above, dataused to train a finite rank deep kernel learning model. Live data 1124may be data provided, for example, by an application, which is to beacted upon by the finite rank deep kernel leaning model. Modelparameters 1126 may be parameters related to, for example, the deepneural network used to determine embeddings, as described above.

While not depicted in FIG. 11, other aspects may be included in storage1110.

As with memory 1110, a single storage 1120 is depicted in FIG. 11 forsimplicity, but the various aspects stored in storage 1120 may be storedin different physical storages, but all accessible to CPU 1102 viainternal data connections, such as bus 1130, or external connection,such as network interface 1108.

The preceding description is provided to enable any person skilled inthe art to practice the various embodiments described herein. Theexamples discussed herein are not limiting of the scope, applicability,or embodiments set forth in the claims. Various modifications to theseembodiments will be readily apparent to those skilled in the art, andthe generic principles defined herein may be applied to otherembodiments. For example, changes may be made in the function andarrangement of elements discussed without departing from the scope ofthe disclosure. Various examples may omit, substitute, or add variousprocedures or components as appropriate. For instance, the methodsdescribed may be performed in an order different from that described,and various steps may be added, omitted, or combined. Also, featuresdescribed with respect to some examples may be combined in some otherexamples. For example, an apparatus may be implemented or a method maybe practiced using any number of the aspects set forth herein. Inaddition, the scope of the disclosure is intended to cover such anapparatus or method that is practiced using other structure,functionality, or structure and functionality in addition to, or otherthan, the various aspects of the disclosure set forth herein. It shouldbe understood that any aspect of the disclosure disclosed herein may beembodied by one or more elements of a claim.

As used herein, the word “exemplary” means “serving as an example,instance, or illustration.” Any aspect described herein as “exemplary”is not necessarily to be construed as preferred or advantageous overother aspects.

As used herein, a phrase referring to “at least one of” a list of itemsrefers to any combination of those items, including single members. Asan example, “at least one of: a, b, or c” is intended to cover a, b, c,a-b, a-c, b-c, and a-b-c, as well as any combination with multiples ofthe same element (e.g., a-a, a-a-a, a-a-b, a-a-c, a-b-b, a-c-c, b-b,b-b-b, b-b-c, c-c, and c-c-c or any other ordering of a, b, and c).

As used herein, the term “determining” encompasses a wide variety ofactions. For example, “determining” may include calculating, computing,processing, deriving, investigating, looking up (e.g., looking up in atable, a database or another data structure), ascertaining and the like.Also, “determining” may include receiving (e.g., receiving information),accessing (e.g., accessing data in a memory) and the like. Also,“determining” may include resolving, selecting, choosing, establishingand the like.

The methods disclosed herein comprise one or more steps or actions forachieving the methods. The method steps and/or actions may beinterchanged with one another without departing from the scope of theclaims. In other words, unless a specific order of steps or actions isspecified, the order and/or use of specific steps and/or actions may bemodified without departing from the scope of the claims. Further, thevarious operations of methods described above may be performed by anysuitable means capable of performing the corresponding functions. Themeans may include various hardware and/or software component(s) and/ormodule(s), including, but not limited to a circuit, an applicationspecific integrated circuit (ASIC), or processor. Generally, where thereare operations illustrated in figures, those operations may havecorresponding counterpart means-plus-function components with similarnumbering.

The various illustrative logical blocks, modules and circuits describedin connection with the present disclosure may be implemented orperformed with a general purpose processor, a digital signal processor(DSP), an application specific integrated circuit (ASIC), a fieldprogrammable gate array (FPGA) or other programmable logic device (PLD),discrete gate or transistor logic, discrete hardware components, or anycombination thereof designed to perform the functions described herein.A general-purpose processor may be a microprocessor, but in thealternative, the processor may be any commercially available processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

A processing system may be implemented with a bus architecture. The busmay include any number of interconnecting buses and bridges depending onthe specific application of the processing system and the overall designconstraints. The bus may link together various circuits including aprocessor, machine-readable media, and input/output devices, amongothers. A user interface (e.g., keypad, display, mouse, joystick, etc.)may also be connected to the bus. The bus may also link various othercircuits such as timing sources, peripherals, voltage regulators, powermanagement circuits, and other circuit elements that are well known inthe art, and therefore, will not be described any further. The processormay be implemented with one or more general-purpose and/orspecial-purpose processors. Examples include microprocessors,microcontrollers, DSP processors, and other circuitry that can executesoftware. Those skilled in the art will recognize how best to implementthe described functionality for the processing system depending on theparticular application and the overall design constraints imposed on theoverall system.

If implemented in software, the functions may be stored or transmittedover as one or more instructions or code on a computer-readable medium.Software shall be construed broadly to mean instructions, data, or anycombination thereof, whether referred to as software, firmware,middleware, microcode, hardware description language, or otherwise.Computer-readable media include both computer storage media andcommunication media, such as any medium that facilitates transfer of acomputer program from one place to another. The processor may beresponsible for managing the bus and general processing, including theexecution of software modules stored on the computer-readable storagemedia. A computer-readable storage medium may be coupled to a processorsuch that the processor can read information from, and write informationto, the storage medium. In the alternative, the storage medium may beintegral to the processor. By way of example, the computer-readablemedia may include a transmission line, a carrier wave modulated by data,and/or a computer readable storage medium with instructions storedthereon separate from the wireless node, all of which may be accessed bythe processor through the bus interface. Alternatively, or in addition,the computer-readable media, or any portion thereof, may be integratedinto the processor, such as the case may be with cache and/or generalregister files. Examples of machine-readable storage media may include,by way of example, RAM (Random Access Memory), flash memory, ROM (ReadOnly Memory), PROM (Programmable Read-Only Memory), EPROM (ErasableProgrammable Read-Only Memory), EEPROM (Electrically ErasableProgrammable Read-Only Memory), registers, magnetic disks, opticaldisks, hard drives, or any other suitable storage medium, or anycombination thereof. The machine-readable media may be embodied in acomputer-program product.

A software module may comprise a single instruction, or manyinstructions, and may be distributed over several different codesegments, among different programs, and across multiple storage media.The computer-readable media may comprise a number of software modules.The software modules include instructions that, when executed by anapparatus such as a processor, cause the processing system to performvarious functions. The software modules may include a transmissionmodule and a receiving module. Each software module may reside in asingle storage device or be distributed across multiple storage devices.By way of example, a software module may be loaded into RAM from a harddrive when a triggering event occurs. During execution of the softwaremodule, the processor may load some of the instructions into cache toincrease access speed. One or more cache lines may then be loaded into ageneral register file for execution by the processor. When referring tothe functionality of a software module, it will be understood that suchfunctionality is implemented by the processor when executinginstructions from that software module.

The following claims are not intended to be limited to the embodimentsshown herein, but are to be accorded the full scope consistent with thelanguage of the claims. Within a claim, reference to an element in thesingular is not intended to mean “one and only one” unless specificallyso stated, but rather “one or more.” Unless specifically statedotherwise, the term “some” refers to one or more. No claim element is tobe construed under the provisions of 35 U.S.C. § 112(f) unless theelement is expressly recited using the phrase “means for” or, in thecase of a method claim, the element is recited using the phrase “stepfor.” All structural and functional equivalents to the elements of thevarious aspects described throughout this disclosure that are known orlater come to be known to those of ordinary skill in the art areexpressly incorporated herein by reference and are intended to beencompassed by the claims. Moreover, nothing disclosed herein isintended to be dedicated to the public regardless of whether suchdisclosure is explicitly recited in the claims.

What is claimed is:
 1. A finite rank deep kernel learning method, comprising: receiving a training dataset; training a deep neural network based on the training data set; forming a set of embeddings based on the trained deep neural network; optimizing an orthogonality of the set of embeddings based on a cost function, the cost function including a penalty term associated with the orthogonality of the set of embeddings; forming, from the set of embeddings, a plurality of dot kernels, wherein the plurality of dot kernels consists of a finite number of dot kernels; combining the plurality of dot kernels to form a composite kernel for a Gaussian process; receiving live data from an application; and predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel.
 2. The method of claim 1, wherein: the composite kernel for the Gaussian process is modeled as a linear combination of the plurality of dot kernels as: K(x,y)=Σ_(i=1) ^(R) ϕ_(i)(x,w)ϕ_(i)(y,w), wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ is an embedding, R is a rank of a combined embedding, and Σ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))² is minimized as a cost function to maintain an orthogonality of the set of embeddings when forming the set of embeddings.
 3. The method of claim 1, wherein the cost function=−log p(y|x)+λΣ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))², wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ is an embedding, and λ is a penalty term associated with the orthogonality of the set of embeddings.
 4. The method of claim 1, wherein the composite kernel for the Gaussian process comprises a finite rank Mercer kernel.
 5. The method of claim 1, wherein: the live data comprises financial data, the application is a financial management application, the plurality of values comprises a plurality of predicted future financial transactions, and each uncertainty of the plurality of uncertainties associated with a respective predicted future financial transaction estimates a range of values of the respective predicted future transaction.
 6. The method of claim 1, wherein: the live data comprises resource utilization data, the application is a resource management application, the plurality of values comprises a plurality of predicted resources needs, and each uncertainty of the plurality of uncertainties associated with a respective predicted future resource need estimates a range of values of the respective resource need.
 7. The method of claim 1, wherein: the live data is user activity data, the application is a resource access control application, the plurality of values comprises a plurality of predicted user activities, and each uncertainty of the plurality of uncertainties associated with a respective predicted future user activity estimates a range of values of the respective user activity.
 8. A system, comprising: a memory comprising computer-executable instructions; a processor configured to execute the computer-executable instructions and cause the system to perform a finite rank deep kernel learning method, the method comprising: receiving a training dataset; training a deep neural network based on the training data set; forming a set of embeddings based on the trained deep neural network; optimizing an orthogonality of the set of embeddings based on a cost function, the cost function including a penalty term associated with orthogonality of the set of embeddings; forming, from the set of embeddings, a plurality of dot kernels, wherein the plurality of dot kernels consists of a finite number of dot kernels; combining the plurality of dot kernels to form a composite kernel for a Gaussian process; receiving live data from an application; and predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel.
 9. The system of claim 8, wherein: the composite kernel for the Gaussian process is modeled as a linear combination of the plurality of dot kernels as: K(x,y)=Σ_(i=1) ^(R) ϕ_(i)(x,w)ϕ_(i)(y,w), wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ is an embedding, R is a rank of a combined embedding, and Σ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))² is minimized as a cost function to maintain an orthogonality of the set of embeddings when forming the set of embeddings.
 10. The system of claim 8, wherein the cost function=−log p(y|x)+λΣ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))², wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ is an embedding, and λ is a penalty term associated with the orthogonality of the set of embeddings.
 11. The system of claim 8, wherein the composite kernel for the Gaussian process comprises a finite rank Mercer kernel.
 12. The system of claim 8, wherein: the live data comprises financial data, the application is a financial management application, the plurality of values comprises a plurality of predicted future financial transactions, and each uncertainty of the plurality of uncertainties associated with a respective predicted future financial transaction estimates a range of values of the respective predicted future transaction.
 13. The system of claim 8, wherein: the live data comprises resource utilization data, the application is a resource management application, the plurality of values comprises a plurality of predicted resources needs, and each uncertainty of the plurality of uncertainties associated with a respective predicted future resource need estimates a range of values of the respective resource need.
 14. The system of claim 8, wherein: the live data is user activity data, the application is a resource access control application, the plurality of values comprises a plurality of predicted user activities, and each uncertainty of the plurality of uncertainties associated with a respective predicted future user activity estimates a range of values of the respective user activity.
 15. A non-transitory computer-readable medium comprising instructions that, when executed by a processor of a processing system, cause the processing system to perform a finite rank deep kernel learning method, the method comprising: receiving a training dataset; training a deep neural network based on the training data set; forming a set of embeddings based on the trained deep neural network; optimizing an orthogonality of the set of embeddings based on a cost function, the cost function including a penalty term associated with orthogonality of the set of embeddings; forming, from the set of embeddings, a plurality of dot kernels, wherein the plurality of dot kernels consists of a finite number of dot kernels; combining the plurality of dot kernels to form a composite kernel for a Gaussian process; receiving live data from an application; and predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel.
 16. The non-transitory computer-readable medium of claim 15, wherein: the composite kernel for the Gaussian process is modeled as a linear combination of the plurality of dot kernels as: K(x,y)=Σ_(i=1) ^(R) ϕ_(i)(x,w)ϕ_(i)(y,w), wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ is an embedding, R is a rank of a combined embedding, and Σ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))² is minimized as a cost function to maintain an orthogonality of the set of embeddings when forming the set of embeddings.
 17. The non-transitory computer-readable medium of claim 15, wherein the cost function=−log p(y|x)+λΣ_(i,j,i≠j) (ϕ_(i)(x,w)^(T)ϕ_(j)(x,w))², wherein: x is a feature derived from an input, y is a response variable, w is a weight, ϕ an embedding, and λ a penalty term associated with the orthogonality of the set of embeddings.
 18. The non-transitory computer-readable medium of claim 15, wherein the composite kernel for the Gaussian process comprises a finite rank Mercer kernel.
 19. The method of claim 1, wherein optimizing the orthogonality of the set of embeddings based on the cost function is done in batches.
 20. The system of claim 8, wherein optimizing the orthogonality of the set of embeddings based on the cost function is done in batches. 